C.R.
I don't think your aunt would have agreed to this commitment if she really didn't want to do it. If she says that she does not want to be paid, I wouldn't worry about compensating her with money. Little gifts here and there (her birthday, mother's day, etc) from you and the kids will make her happy. Pictures of the kids, a homemade meal, crafts the kids have made would be nice. I think that her knowing how much you appreciate her would be the only compensation she is looking for.
You are fortunate that you are able to work this out with your family. If I had family that could help out, I would love to do some kind of work.
Good luck at the new job. I have several friends that do that and they love it. Especially when their kids go to school.